The inequality Pr (a≤Wi≤b;i= 1, 2, …,n) ≥ IInc=iPr (a≤Wi≤b) is shown to hold when the random variablesW1, …,Wn, have a particular multivariatef-distribution which is defined. It is used in this paper to construct conservative simultaneous prediction intervals for failure times in certain life test experiments. An additional inequality is shown to hold for all bivariate distributions belonging to a class of bivariatef-distributions which is defined. An indication of the use of this latter inequality is given. Some tables are included which indicate the sharpness of each of the inequalities.